Questions tagged [prime-factorization]

For questions about factoring elements of rings into primes, or about the specific case of factoring natural numbers into primes.

A natural number is prime if it has no positive divisors besides $1$ and itself. The fundamental theorem of arithmetic states that every natural number $n>1$ can be factored uniquely, up to a reordering of the factors, as a product of distinct prime numbers each raised to some power.

This concept holds in a more general setting though. A ring is called a unique factorization domain (UFD) if every non-unit element can be factored uniquely as a product of prime elements in the ring. The ring of integers is an example of a UFD.

1894 questions
13
votes
3 answers

Is there a name/notation for the sum of the powers in a prime factorization

Let $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, where $p_i$'s are distinct primes and $\alpha_i \geq 1$ for all $i$. Is there any name & notation for the number $\alpha_1 + \alpha_2+ \cdots + \alpha_k$?
RKR
  • 517
  • 3
  • 14
13
votes
2 answers

Finding the largest subset of factors of a number whose product is the number itself

Given a positive integer $x$, find $k$ distinct positive integers $y_1, y_2, \dots, y_k$ such that $$ x = \prod_{i=1}^k(1+y_i) $$ The problem is to pick the $y$'s so that $k$ is as large as possible. Now, if the restriction of distinctness of $y$'s…
Priyatham
  • 2,567
  • 14
  • 29
12
votes
1 answer

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5})\cdot(4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two factorizations of $21$ into pairwise non-associate…
deh
  • 123
  • 1
  • 4
12
votes
2 answers

$K[x_1, x_2,\dots ]$ is a UFD

I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field. If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization of $f$ in $R$ only have factors in these…
12
votes
1 answer

An old multiplication technique and its reverse for Integer Factoring

An ancient Indian multiplication technique is as follows: $$\array{ a=107 & +7 & (\text{base}\space r=100)\\ b=113 & +13 \\ \hline (a+b-r)=(107 + 13) & (7 \times 13) & \space\text{or}\\ (b+a-r)=(113 + 7) & (7 \times 13) \\ \hline 120…
vvg
  • 1,075
  • 3
  • 22
12
votes
3 answers

On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two

I don't know if the following diophantine equation (problem) is in the literature. We consider the diophantine equation $$x^{m-1}(x+1)=y^{n-1}(y+1)\tag{1}$$ over integers $x\geq 2$ and $y\geq 2$ with $x>y$, and over integers $m\geq 2$ and $n\geq 2$.…
12
votes
4 answers

What is notable about the composite numbers between twin primes?

Look at the composites between twin primes (A014574): $$ 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, \\ 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, \ldots \;. $$ Is there anything special about their…
Joseph O'Rourke
  • 29,337
  • 4
  • 72
  • 147
12
votes
2 answers

Expression for the highest power of 2 dividing $3^a\left(2b-1\right)-1$

Question: I am wondering if an expression exists for the highest power of 2 that divides $3^a\left(2b-1\right)-1$, in terms of $a$ and $b$, or perhaps a more general expression for the highest power of 2 dividing some even $n$? EDIT: This is…
12
votes
1 answer

$2^i - 2293$ is always composite?

Is $2^i - 2293$ always composite for $i=1,2,3,...$ ? I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$ In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}] Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943, …
a boy
  • 841
  • 4
  • 12
12
votes
2 answers

Consecutive numbers that share the same sum of prime factors

Let $f(n)$ denote the sum of the prime factors of $n$ (with multiplicity). I have been looking for pairs of consecutive numbers $n,n+1$ such that $f(n)=f(n+1)$. Case #$1$: $f(8)=f(2\cdot2\cdot2)=2+2+2=6$ $f(9)=f(3\cdot3)=3+3=6$ Case…
barak manos
  • 42,243
  • 8
  • 51
  • 127
11
votes
4 answers

Find the prime-power decomposition of 999999999999

I'm working on an elementary number theory book for fun and I have come across the following problem: Find the prime-power decomposition of 999,999,999,999 (Note that $101 \mid 1000001$.). Other than just mindlessly guessing primes that divide it,…
candido
  • 625
  • 5
  • 14
11
votes
1 answer

Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
Peter
  • 78,494
  • 15
  • 63
  • 194
11
votes
4 answers

Why perfect square has odd number of factors

can someone please describe me why only the perfect square has odd number of factors.why does other number not has odd numbers of factors? I understand it but don't find any mathmetical proof.Please help me
11
votes
2 answers

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are factors of our number. The way that people can use known…
11
votes
3 answers

Interesting thing about sum of squares of prime factors of $27$ and $16$.

Let $$n=p_1×p_2×p_3×\dots×p_r$$ where $p_i$ are prime factors and $f$ is the functions $$f(n)=p_1^2+p_2^2+\dots+p_r^2$$ If we put $n=27,16$ and $27=3×3×3$, $16=2×2×2×2$…